Page 149 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes finite set. Our example is the set Z,,. Let us briefly recall the construction of Z,, the set of residue
classes modulo n.
If a and b are integers, we say that a is congruent to b modulo n if a – b is divisible by n; in
symbols, a  b (mod n) if n I (a – b). The relation ‘congruence modulo n’ is an equivalence
relation in Z. The equivalence class containing the integer a is

a = { b  Z ( a - b is divisible by n }
= { a + m p | m  Z }.
It is called the congruence class of a modulo n or the residue class of a modulo n. The set of all
equivalence classes is denoted by Z,,. So

Z , , = {0,1,2,...,n 1}.

We define addition and multiplication of classes in terms of their representatives by

a
a  b   b and
a . b  ab  a, b Zn.

To help you regain some practice in adding and multiplying in Z,,, consider the following
Cayley tables for Z .
n

Now let us go back to looking for a finite ring.

Example: Show that (Z , +, .) is a ring.
n
Solution: You already know that (Z , +) is an abelian group, and that multiplication is associative
n
in Z,. Now we need to see if the axiom R6 is satisfied.

For any a, b c Z , n




a.(b c) a.(b c) a.b a.c  a.b a.c  ab ac




Similarly, ( a + 6) . = L.; + b.c  a, b c Z, .
So, (Z , +,) satisfies the axioms R1-R6. Therefore, it is a ring.
n
Now let us look at a ring whose underlying set is a subset of C.
Example: Consider the set
Z + iZ = { m + in | m and n are integers }, where i = – I.
2

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